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  2. Proportional cake-cutting with different entitlements - Wikipedia

    en.wikipedia.org/wiki/Proportional_cake-cutting...

    Cut-near-halves may need at least four cuts: first, George cuts in the ratio 5:5, and Alice gets 5. Then, Alice cuts in the ratio 3:2; suppose George chooses the 2. Then, George cuts in the ratio 2:1; suppose Alice chooses the 1. Finally, they do cut-and-choose on the remainder. We can do better by letting George cut in the ratio 6:4.

  3. Dividing a square into similar rectangles - Wikipedia

    en.wikipedia.org/wiki/Dividing_a_square_into...

    However, there are three distinct ways of partitioning a square into three similar rectangles: [1] [2] The trivial solution given by three congruent rectangles with aspect ratio 3:1. The solution in which two of the three rectangles are congruent and the third one has twice the side length of the other two, where the rectangles have aspect ...

  4. Tiling with rectangles - Wikipedia

    en.wikipedia.org/wiki/Tiling_with_rectangles

    The smallest square that can be cut into (m × n) rectangles, such that all m and n are different integers, is the 11 × 11 square, and the tiling uses five rectangles. [1] The smallest rectangle that can be cut into (m × n) rectangles, such that all m and n are different integers, is the 9 × 13 rectangle, and the tiling uses five rectangles ...

  5. Divide and choose - Wikipedia

    en.wikipedia.org/wiki/Divide_and_choose

    Divide-and-choose might yield inefficient allocations. One commonly used example is a cake that is half vanilla and half chocolate. Suppose Bob likes only chocolate, and Carol only vanilla. If Bob is the cutter and he is unaware of Carol's preference, his safe strategy is to divide the cake so that each half contains an equal amount of chocolate.

  6. Guillotine partition - Wikipedia

    en.wikipedia.org/wiki/Guillotine_partition

    Guillotine partition is the process of partitioning a rectilinear polygon, possibly containing some holes, into rectangles, using only guillotine-cuts. A guillotine-cut (also called an edge-to-edge cut) is a straight bisecting line going from one edge of an existing polygon to the opposite edge, similarly to a paper guillotine.

  7. Wallace–Bolyai–Gerwien theorem - Wikipedia

    en.wikipedia.org/wiki/Wallace–Bolyai–Gerwien...

    By the Wallace–Bolyai–Gerwien theorem, a square can be cut into parts and rearranged into a triangle of equal area. In geometry , the Wallace–Bolyai–Gerwien theorem , [ 1 ] named after William Wallace , Farkas Bolyai and P. Gerwien , is a theorem related to dissections of polygons .

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  9. Centroid - Wikipedia

    en.wikipedia.org/wiki/Centroid

    Divide the shape into two rectangles, as shown in fig 2. Find the centroids of these two rectangles by drawing the diagonals. Draw a line joining the centroids. The centroid of the shape must lie on this line . Divide the shape into two other rectangles, as shown in fig 3.

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